# Coefficient of Variation

Coefficient of variation is a measure used to assess the total risk per unit of return of an investment. It is calculated by dividing the standard deviation of an investment by its expected rate of return.

Since most investors are risk-averse, they want to minimize their risk per unit of return. Coefficient of variation provides a standardized measure of comparing risk and return of different investments. A rational investor would select an investment with lowest coefficient of variation.

Sharpe ratio is a similar statistic which measures excess return per unit of risk.

## Formula

$$\text{Coefficient of Variation} \\ = \frac{\text{Standard Deviation of the Investment}}{\text{Expected Return on the Investment}}$$

## Example

Indus Farms is a family owned business engaged in cultivating their land mass of a hundred square kilometers. The season is beginning, and Akbar the family head, has a critical decision to make: to cultivate sugar cane or cotton. He tasked his eldest son Adnan to gather some data on expected return on each crop under different scenarios and the variation in those returns.

Adnan estimates that if there are enough rains (which has a probability of 0.7), the return on sugar cane could be as high as 25%. However, in case of low rain, the return could be as low as 5%. He estimates that standard deviation of return on sugar cane crop is 14%. In case of enough rains, return on cotton could be only 12%, but in case of low rain, the return could be 20%. Standard deviation of return on cotton is expected to be 9%. In the risk-return perspective, which crop is better for Akbar?

Expected return on sugar cane = 0.7 × 25% + 0.3 × 5% = 19%

$$\text{Coefficient of Variation of Sugar Cane Cultivation} \\ = \frac{\text{Standard Deviation on Sugar Cane}}{\text{Expected Return on Sugar Cane}} \\ = \frac{\text{16%}}{\text{19%}} \\ = \text{0.74}$$

Expected return on cotton = 0.7 × 12% + 0.3 × 20% = 14.4%

$$\text{Coefficient of Variation of Cotton Cultivation} \\ = \frac{\text{Standard Deviation on Cotton}}{\text{Expected Return on Cotton}} \\ = \frac{\text{9%}}{\text{14.4%}} \\ = \text{0.625}$$

Since cotton cultivation has the lower coefficient of variation, it offers less risk per unit of return. Akbar should prefer cotton over sugar cane.